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Having never been a mathematics educator, my question could be stupid and, if this is the case, please delete it.

When I was young, from the very beginning of limits, we were teached that there are two things to look at simultaneously : what is the limit and how it is approached. This means that, even when quite young, the concept of asymptotics was in our minds and, thinking back, I have the personal feeling that this was a great idea.

Visiting almost daily MSE, I have the feeling that this is not the case almost anywhere.

So my question : am I totally out of date ? Should I be wrong if I tried to teach that way ?

Claude Leibovici
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  • See my answer at http://matheducators.stackexchange.com/a/10077/1550, which contains an explanation of this idea in a way that can be taught to high-school students, but sadly everyone who has read it either don't appreciate it or don't even understand it... – user21820 Nov 26 '15 at 08:31
  • Thanks for answering. Your answer is very interesting. In fact, not being an educator, what I find funny is that there is no conceptual problem when trying to teach asymptotics and/or Taylor expansions. And, from there, everything becomes so simple when we have to deal with limits ! – Claude Leibovici Nov 26 '15 at 08:40
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    Exactly. Perhaps as a non-educator you have a better understanding of what students (who are also non-educators) face in understanding limits! =) – user21820 Nov 26 '15 at 08:41
  • And I should add that asymptotic analysis is so very natural because it follows straight from generalizing linear approximations (which occur for all differentiable curves). – user21820 Nov 26 '15 at 08:43
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    I don't know how young pupils/students percieve mathematics outside France. Here, they consider that it is a punishment to suffer and that it is of no use. When I show that mathematics can lead to beauty and even fun, most of the time, I suppose they look at me just as if I was coming from Mars. – Claude Leibovici Nov 26 '15 at 08:47
  • @user21820. Clearly, we agree on almost everything ! I really appreciate. By the way, where are you ? – Claude Leibovici Nov 26 '15 at 08:48
  • I was recently browsing my old chat-rooms and realized that perhaps you could not access this chat-room that I'd created for us to continue our discussion. You're welcome to come to the logic chat-room to chat anytime! Also, concerning teaching asymptotic analysis to students, you may be interested in looking at my answer here recently in case you didn't see it. =) – user21820 Apr 06 '17 at 16:10
  • @user21820. Thanks for your message and the links. I should be delighted to continue on chat. Whenever you want, open a room and tell it to me on Mathematics SE or e-mail (my address is in my profile). I am almot available at any time from 6:00am to 1:00pm (French time). Cheers. – Claude Leibovici Apr 07 '17 at 03:35

2 Answers2

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If I understand you, I do often think this way when solving a problem. For example, see this answer of mine at Mathematics Stack Exchange.

However, limits involving infinity (as the independent or dependent variable) seem to be losing importance in textbooks. For example, I teach from Calculus: Graphical, Numerical, Algebraic by Ross L. Finney et al., and that book sticks "Limits Involving Infinity" into one section and largely ignores the concept otherwise. This approach has some advantages, such as not needing to distinguish between a limit of plus-or-minus infinity and an undefined limit. But you see it has some disadvantages as well.

Community
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Rory Daulton
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  • I know you for quite a while and, from answers at MSE, I think that we share many points of view on problems. Thanks for taking the time of answering. – Claude Leibovici Jul 01 '15 at 10:36
  • I feel the idea of focusing at first only on finite limits (at a finite value of $x$) is justified on pedagogical grounds. The reason is that students are bored with limits in the initial stages, and invariably all we do in studying limits initially is to rewrite expressions in a different form that makes the limit obvious by substitution. It is difficult for students to understand what the substance is in this formalism. The idea of this approach is to restrict the study of limits to the bare minimum needed to define and compute derivatives, returning to subtler questions... – Keith Jul 02 '15 at 01:00
  • (cont'd) including infinite limits and limits at infinity, in the context of curve sketching. – Keith Jul 02 '15 at 01:01
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I'd recommend not being intimidated by textbook trends! "In real life" (as opposed to textbook-life, for sure, and often opposed to required-curriculum "school-math"), _of_course_ the two things you mention, "the limit", and "how it is approached", matter a great deal. Do not be intimidated by silly books (written by non-mathematicians, almost entirely) to think otherwise. That is, _of_course_ we care about the asymptotics as you mention. True, "sadly" for immature students, those details are more complicated than the more-typical (in the U.S.) symbol-pushing, and it would be hard to convince the kids to pay attention at all if "it's not on the final", ... which it cannot be, mostly, because it is not the general standard.

But, yes, there is an underlying reality of human understanding of these things, in which asymptotics play a large role...

paul garrett
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